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 A242861 Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0. 11
 1, 1, 1, 4, 2, 1, 12, 44, 56, 18, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 40, 686, 6632, 39979, 157000, 407620, 695848, 762180, 510752, 192672, 35104, 2180, 1, 60, 1622, 26172, 281514, 2135356, 11785382, 48145820, 146702793, 333518324, 562203148 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also, coefficients of the matching-generating polynomial of the n X n grid graph. In the n-th row there are floor(n^2/2)+1 values. LINKS Alois P. Heinz, Rows n = 0..14, flattened Ralf Stephan, Two dominoes on the 3x3 chessboard, illustration of T(3,2)=44. Eric Weisstein's World of Mathematics, Grid Graph Eric Weisstein's World of Mathematics, Matching-Generating Polynomial Wikipedia, Matching polynomial Index entries for sequences related to dominoes Index entries for sequences related to matchings FORMULA T(n,1) = A046092(n-1), T(n,2) = A242856(n). T(n,floor(n^2/2)) = A137308(n), T(2n,2n^2) = A004003(n). sum(k>=0, T(n,k)) = A210662(n,n) = A028420(n). T(n,3) = A243206(n), T(n,4) = A243215(n), T(n,5) = A243217(n), T(n,floor(n^2/4)) = A243221(n). - Alois P. Heinz, Jun 01 2014 EXAMPLE Triangle starts: 1 1 1 4 2 1 12 44 56 18 1 24 224 1044 2593 3388 2150 552 36 1 40 686 6632 39979 157000 407620 695848 762180 510752 192672 35104 2180 ... MAPLE b:= proc(n, l) option remember; local k; if n=0 then 1 elif min(l[])>0 then b(n-1, map(h->h-1, l)) else for k while l[k]>0 do od; expand(`if`(n>1, x*b(n, subsop(k=2, l)), 0) +`if`(k (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0\$n])): seq(T(n), n=0..8); # Alois P. Heinz, Jun 01 2014 MATHEMATICA b[n_, l_List] := b[n, l] = Module[{k}, Which[n == 0, 1, Min[l]>0, b[n-1, l-1], True, For[k=1, l[[k]]>0, k++]; Expand[If[n>1, x*b[n, ReplacePart[l, k -> 2]], 0] + If[k 1, k + 1 -> 1}]], 0] + b[n, ReplacePart[l, k -> 1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, Array[0&, n]]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 16 2015, after Alois P. Heinz *) PROG (Sage) def T(n, k): G = Graph(graphs.Grid2dGraph(n, n)) G.relabel() mu = G.matching_polynomial() return abs(mu[n^2-2*k]) CROSSREFS Cf. A046741, A096713. Sequence in context: A174005 A152818 A302235 * A109244 A171650 A225476 Adjacent sequences: A242858 A242859 A242860 * A242862 A242863 A242864 KEYWORD nonn,tabf AUTHOR Ralf Stephan, May 24 2014 STATUS approved

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Last modified August 14 10:04 EDT 2024. Contains 375159 sequences. (Running on oeis4.)